Question: $g'(x)=\dfrac{2x+5y}{x}$ Is $g(x)=x^5-2x$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
In order to find whether $g(x)=x^5-2x$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $g(x)$, we need to find the corresponding $g'(x)$ expression to substitute into the equation: $\begin{aligned} g'(x)&=\dfrac{d}{dx}\left[x^5-2x\right] \\\\ &=5x^4-2 \end{aligned}$ Now we substitute ${g(x)=x^5-2x}$ and ${g'(x)=5x^4-2}$ into the equation: $\begin{aligned} {g'(x)}&=\dfrac{2x+5{g(x)}}{x} \\\\ {5x^4-2}&\stackrel{?}{=}\dfrac{2x+5\left({x^5-2x}\right)}{x} \\\\ 5x^4-2&\stackrel{?}{=}\dfrac{2x+5x^5-10x}{x} \\\\ 5x^4-2&\stackrel{?}{=}\dfrac{5x^5-8x}{x} \\\\ 5x^4-2&\stackrel{?}{=} 5x^4-8 \\\\ -2&\neq -8 \end{aligned}$ We did not obtain equivalent expressions on each side. In conclusion, no, $g(x)=x^5-2x$ is not a solution to the differential equation.